Market forces meet behavioral biases: cost misallocation and irrational pricing.

* In this section, we examine the important special case of sym metric linear demand. This additional structure enables us to generate comparative statics predictions about how the distortion changes with the number of firms, the degree of product differentiation, and so on. Theorems 1 and 2 illustrated that our main points hold generally. Once we move to the special structure of symmetric linear demand, the analysis becomes much simpler: there is a unique equilibrium in the pricing game for any profile of distortions. The cost distortion game is supermodular, dominance solvable, and its Nash equilibrium is also the unique correlated equilibrium (Milgrom and Roberts, 1990). A broad class of learning models, including our own, converge to the unique equilibrium in the distortion game.

We assume that the N firms in the industry have a common marginal cost c and face a symmetric system of firm-level demand curves that is consistent with maximization by a representative consumer who has a quadratic net benefit function. The representative consumer chooses quantities q = ([q.sub.1], ... [q.sub.N]) to maximize

U(q) = aq - 1 / 2 bq [THETA]q - pq,

where a and b are positive constants and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [theta] [member of] (0, 1) parameterizes the extent of horizontal differentiation among the goods. As [theta] [right arrow] 0, the goods become independent, and as [theta] [right arrow] 1, the goods become perfect substitutes. Throughout, we assume that a > c.

The system of demand functions implied by the solution to the representative consumer's utility maximization problem is given by

[q.sub.n]= [D.sub.n](p) = [alpha] - [beta][p.sub.n] + [gamma] [summation over (m[not equal to]n)] [p.sub.m], n [member of] {1, ..., N},

where

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For later use, note that

2[beta] - (N - 1)[gamma] = [2(1 - [theta]) + (N - 1)[theta]/b(1 - [theta])[1 + (N - 1)[theta]]] > 0.

The (unique, symmetric) Nash equilibrium in prices without distortions is

[p.sup.0] = [lambda]a + (1 - [lambda])c,

where

[lambda] = [(1 - [theta]) / 2(1 - [theta]) + (N - 1)[theta]] [member of] (0, 1/2).

Now, suppose that firm n chooses costing methodology [s.sub.n], [member of] [0, [s.sup.+]], where [s.sup.+] [equivalent to] F/[??] is the maximum distortion consistent with a firm's sunk cost and is assumed to be common across all firms. The first-order condition for a firm is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Given a vector s of distortions, firm n's second-stage equilibrium price [[bar.p].sub.n] (s) is found by solving this system of first-order conditions:

[[bar.p].sub.n](s) = [p.sup.0] + [[beta]/2[beta] + [gamma]][s.sub.n] + [[beta]/2[beta]+[gamma]]([[gamma]/2[beta] - (N - 1)[gamma]])[N.summation over (m=1)][s.sub.m]. (4)

This implies

[partial derivative]/[[bar.p].sub.n]/[partial derivative]/[s.sub.n] = [[beta]/2[beta] + [gamma]] [1 + [[gamma]/2[beta] - (N - 1)[gamma]]] > 0. (5)

[partial derivative]/[[bar.p].sub.n]/[partial derivative]/[s.sub.m] = [[beta]/2[beta] + [gamma]] [[gamma]/2[beta] - (N - 1)[gamma]] > 0. (6)

Finally, firm n's economic profit is

[[bar.[tau]].sup.e.sub.n](s) [equivalent to] [[bar.[tau]].sup.e.sub.n]([[bar.p].sub.n](s)) = [[bar.p].sub.n](s) - c) [[alpha] - [beta][[bar.p].sub.n](s) + [gamma] [summation over (m[not equal to]n)[[bar.p].sub.m](s)]. (7)

We call the game where the firms choose distortions and payoffs are determined by (7) and (4) the distortion game.

Firm n's problem in the distortion game is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We begin by establishing some basic properties of the distortion game:

Proposition 3. The distortion game is supermodular and has a unique, symmetric equilibrium distortion [s.sup.*].

Throughout, we assume that the symmetric equilibrium in the first stage is less than the upper bound [s.sup.+], and below we show that this distortion is generally positive for [theta] [member of] (0, 1 ). Given this, the equilibrium distortion [s.sup.*] satisfies

[partial derivative][bar.[tau]].sup.e.sub.n]([s.sup.*])/[[partial derivative][s.sub.n] = 0, n [member of] {1, ..., N},

with the induced equilibrium price [p.sup.*] = [[bar.p].sub.n] ([s.sup.*]), n [member of] {1, ..., N}. These conditions can be shown to imply

[s.sup.*] = [([p.sup.*] - c)(N - 1)[xi][gamma]/[beta]] (8)

[p.sup.*] = [p.sup.0] + [[beta][s.sup.*] / 2[beta] - (N - 1)[gamma]], (9)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With this derivation in hand, we can establish a baseline set of no-distortion results that mirror those in Section 4:

Proposition 4. (i) As the goods become independent, the equilibrium distortion goes to zero, that is, [lim.sub.[theta][right arrow]0] [s.sup.*] = 0; (ii) as the goods become perfect substitutes, the equilibrium distortion goes to zero, that is, [lim.sub.[theta][right arrow]0] [s.sup.*] = 0; (iii) as the number of firms becomes infinitely large, the equilibrium distortion goes to zero, i.e., [lim.sub.N[right arrow][infinity]] that is [s.sup.*] = 0.

We now turn our attention to circumstances under which the equilibrium distortion is positive. To do so, we solve (8) and (9) in terms of the primitives of the model: a, b, c, [theta], and N.

[p.sup.*] - c = [([p.sup.0] - c)[(2 - [theta]) + (N - 1)[theta]] / (2 - [theta]) + (N - 1)[theta](1 - [theta])] (10)

[s.sup.*] = [([p.sup.0] - c)(N - 1)[[theta].sup.2] / [(2 - [theta]) + (N - 1)[theta](1 - [theta])][1 - [theta]) + (N - 1)[theta]]]. (11)

An immediate implication of these expressions is that when there is some degree of (imperfect) product differentiation, firms distort their relevant costs in equilibrium, which in turn elevates the equilibrium price relative to the single-stage Nash equilibrium.

Proposition 5. If [theta] [member of] (0, 1), the equilibrium distortion is positive, that is, [s.sup.*] > 0. Moreover, the induced equilibrium price exceeds the Nash equilibrium price without distortions, that is, [p.sup.*] > [p.sup.0.]

Table 1 shows calculations of the Nash price, the equilibrium distortion [s.sup.*], and the induced equilibrium price [p.sup.*] for different numbers N of firms and values of the product differentiation parameter, [theta]. (22) Note that the equilibrium distortion is not monotone in N. For low values of [theta], the distortion initially increases in N, but eventually decreases to 0 as N becomes sufficiently large. As one would expect given Proposition 4, the equilibrium distortion is not monotone in [theta], either. For any N, it initially increases in [theta] but eventually begins to decrease, going to zero as the products become perfect substitutes. Thus, for near-perfectly homogeneous products, the cost distortion is very small. There is indirect evidence to support this latter conclusion: Maher, Stickney, and Weilt 2004) note that "cost-based pricing is far less prevalent in Japanese process-type industries (for example, chemicals, oil, and steel)," all of which are products with very weak differentiation. By contrast, in an industry with a small number of firms and an intermediate degree of product differentiation, the equilibrium distortion can be significant. For instance, when N = 2 and [theta] = 0.80, the equilibrium distortion is $21.82, about 36% of the undistorted Nash price of $60. Table 1 suggests that even though the distortion is nonmonotonic in the number of firms, the induced equilibrium price is. The next proposition confirms this.

Proposition 6. The induced equilibrium price [p.sup.*] is strictly decreasing in the number of firms.

Because firms do not internalize the beneficial impact of their distortion of relevant costs on their rivals' profits, they end up with a price lower than the monopoly price.

Proposition 7. When [theta] [member of] (0, 1), the induced equilibrium price is less than the monopoly price.

Let us now turn to a comparative statics analysis of the equilibrium distortions with respect to the other parameters of the model, a and c. To avoid conflating the impact of a and c on the distortion with their effect on the overall price level, we derive the comparative statics result for the percentage distortion above the undistorted Nash equilibrium price, [s.sup.*]/[p.sup.0].

Proposition 8. The percentage distortion [s.sup.*]/[p.sup.0] is increasing in a and decreasing in c. Hence an increase in demand, as measured by a larger value of a, will result in a greater percentage distortion, whereas an increase in marginal cost will result in a lower-percentage distortion.

This result makes sense. The difference a - c measures the intrinsic value of the industry profit opportunity. The incremental benefit to a firm from suppressing price competition will be greater the more intrinsically profitable the market is, and, as a result, the degree of distortion is greater the greater is a or the lower is c.

As a final point, consider the implications of this analysis for entry. Because distortions increase equilibrium margins, one would expect that entry would be more desirable in that case, and more firms would come into the industry. We can confirm this intuition. Imagine that each firm faces a sunk cost of entry F, and let [N.sup.0] and [N.sup.*] be the equilibrium number of firms in the no-distortion and distortion cases, respectively. Given the equilibrium number of firms, economic profit is zero, which implies

(a - [p.sup.0]/b) ([p.sup.0] - c]/1 + ([N.sup.0] - 1)[theta]] = F (12)